56.36.2 problem Ex 2

Internal problem ID [14281]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 2
Date solved : Thursday, October 02, 2025 at 09:27:40 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+y&=x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+y(x) = x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2}{x}+\frac {x}{4}+\frac {c_1 \ln \left (x \right )}{x} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 26

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+y[x]==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2+4 c_2 \log (x)+4 c_1}{4 x} \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) - x + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + C_{2} \log {\left (x \right )} + \frac {x^{2}}{4}}{x} \]